Self-regulating genes. Exact steady state solution by using Poisson representation

Sugár, István P.; Simon, István

doi:10.2478/s11534-014-0497-0

Cited by 7 publications

(7 citation statements)

References 10 publications

(38 reference statements)

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“…(iii) The chemical master equation for each model admits an exact solution in steady-state conditions. These exact solutions have been obtained using the method of generating functions but in other studies using similar models, the solution was obtained using the Poisson representation [10,23,24,25]. There are however important differences between the models particularly how they describe protein production and protein-gene interactions that are not often spelled out but can be discerned from the form of the CME.…”

Section: Discrete Models Of Auto-regulationmentioning

confidence: 99%

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Holehouse

Cao

Grima

2020

Biophysical Journal

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Auto-regulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to (i) which sub-cellular processes are explicitly modelled; (ii) the modelling methodology employed (discrete, continuous or hybrid); (iii) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models in the literature, summarize our current understanding of the relationship between them and highlight some of the insights gained through modelling. Discrete Models of auto-regulationFrom a biologist perspective, a minimal model of auto-regulation should describe the main biochemical processes describing the flow of information from gene to mRNA to protein and back to the gene. Hence the model should describe transcription and translation (the two steps at the heart of the central dogma of molecular biology), mRNA and protein degradation, and interactions of proteins with genes. For simplicity we consider the case where there is a single gene copy and all processes are modelled as first-order reactions except the protein-gene interactions which naturally follow second-order kinetics.We refer to this model as the full model since it will be our ground truth, i.e. the finest scale model that we shall consider here. The reactions describing this model are GWhile this model is intuitive, it has not been studied extensively because the mathematical description of its stochastic dynamics, as provided by the chemical master equation, is not easy to solve analytically. In fact even in the absence of the feedback loop, i.e. no protein-gene interactions, its master equation has still not been solved exactly [17]. Hence historically, simplified versions of the full model have received much more attention in the literature. These are the models by Hornos et al. [18] in 2005, Grima et al. [19] in 2012 and Kumar et al. [20] in 2014. Henceforth we shall refer to these as the Hornos, Grima and Kumar models. There exist other discrete models e.g. [21] which in certain limits reduce to the aforementioned three.The three models share a few common properties: (i) They only describe protein fluctuations, i.e. there is no explicit mRNA description. (ii) The models are discrete in the sense that protein numbers change by discrete integer amounts when reactions occur. (iii) The chemical master equation for each model admits an exact solution in steady-state conditions. These exact solutions have been obtained using the method of generating functions but in other studies using similar models, the solution was obtained using the Poisson representation [10,23,24,25]. There are however important differences between the models particularly how they describe protein production and protein-gene interactions that are not often spelled out but can be disc...

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Section: Discrete Models Of Auto-regulationmentioning

confidence: 99%

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Holehouse

Cao

Grima

2020

Biophysical Journal

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“…Our approach is based on the Poisson representation, initially introduced by Gardiner and Chaturvedi [15] as a powerful ansatz-based technique for solving master equations. As emphasized by the authors, this representation is particularly adapted to chemical birth-death processes because of the particular jump rate form implied by stochastic mass-action kinetics [15,26], and it has already been successfully applied to intracellular kinetics [36] and more specifically to gene expression [20,35,29]. In our case, contrary to the original approach [15] in which all species are included, and as also done tacitly in [20,35], we shall apply the Poisson representation only to the mRNA part (species M) and not to the promoter part (species S i , i ∈ 1, n ).…”

Section: Poisson Representationmentioning

confidence: 99%

“…As emphasized by the authors, this representation is particularly adapted to chemical birth-death processes because of the particular jump rate form implied by stochastic mass-action kinetics [15,26], and it has already been successfully applied to intracellular kinetics [36] and more specifically to gene expression [20,35,29]. In our case, contrary to the original approach [15] in which all species are included, and as also done tacitly in [20,35], we shall apply the Poisson representation only to the mRNA part (species M) and not to the promoter part (species S i , i ∈ 1, n ). The representation then becomes more than just an ansatz by revealing an actual "hidden layer" that turns out to be a piecewise-deterministic Markov process.…”

Section: Poisson Representationmentioning

confidence: 99%

Stochastic Gene Expression with a Multistate Promoter: Breaking Down Exact Distributions

Herbach¹

2019

SIAM J. Appl. Math.

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We consider a stochastic model of gene expression in which transcription depends on a multistate promoter, including the famous two-state model and refractory promoters as special cases, and focus on deriving the exact stationary distribution. Building upon several successful approaches, we present a more unified viewpoint that enables us to simplify and generalize existing results. In particular, the original jump process is deeply related to a multivariate piecewisedeterministic Markov process that may also be of interest beyond the biological field. In a very particular case of promoter configuration, this underlying process is shown to have a simple Dirichlet stationary distribution. In the general case, the corresponding marginal distributions extend the well-known class of Beta products, involving complex parameters that directly relate to spectral properties of the promoter transition matrix. Finally, we illustrate these results with biologically plausible examples.

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“…The Poisson representation was first introduced by Gardiner and Chaturvedi [11–13] and serves as a powerful tool to investigate the properties of chemical master equations. In particular, it has been used to solve the chemical master equations of some important gene expression models [14–16]. Recently, some studies [17–19] used the Poisson representation to examine the relationship between discrete and continuous gene expression models and found that the gene product number distributions of some simple discrete and continuous models are related by the Poisson representation.…”

Section: Introductionmentioning

confidence: 99%

Poisson representation: a bridge between discrete and continuous models of stochastic gene regulatory networks

Wang,

Li,

Jia

2023

J. R. Soc. Interface.

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Stochastic gene expression dynamics can be modelled either discretely or continuously. Previous studies have shown that the mRNA or protein number distributions of some simple discrete and continuous gene expression models are related by Gardiner’s Poisson representation. Here, we systematically investigate the Poisson representation in complex stochastic gene regulatory networks. We show that when the gene of interest is unregulated, the discrete and continuous descriptions of stochastic gene expression are always related by the Poisson representation, no matter how complex the model is. This generalizes the results obtained in Dattani & Barahona (Dattani & Barahona 2017 J. R. Soc. Interface 14 , 20160833 ( doi:10.1098/rsif.2016.0833 )). In addition, using a simple counter-example, we find that the Poisson representation in general fails to link the two descriptions when the gene is regulated. However, for a general stochastic gene regulatory network, we demonstrate that the discrete and continuous models are approximately related by the Poisson representation in the limit of large protein numbers. These theoretical results are further applied to analytically solve many complex gene expression models whose exact distributions are previously unknown.

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Self-regulating genes. Exact steady state solution by using Poisson representation

Cited by 7 publications

References 10 publications

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study

Stochastic Gene Expression with a Multistate Promoter: Breaking Down Exact Distributions

Poisson representation: a bridge between discrete and continuous models of stochastic gene regulatory networks

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